On Equitable Coloring of Book Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

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Classes of graphs with restricted interval models

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.263 ◽

1999 ◽

Vol Vol. 3 no. 4 ◽

Author(s):

Andrzej Proskurowski ◽

Jan Arne Telle

Keyword(s):

Maximum Clique ◽

Interval Graphs ◽

Induced Subgraph ◽

Graph Theoretic ◽

Graph Classes ◽

International Audience ◽

Forbidden Induced Subgraph ◽

Nested Hierarchy ◽

Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

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Lower Bounds for Embedding into Distributions over Excluded Minor Graph Families

Algorithms – ESA 2004 - Lecture Notes in Computer Science ◽

10.1007/978-3-540-30140-0_15 ◽

2004 ◽

pp. 146-156 ◽

Cited By ~ 1

Author(s):

Douglas E. Carroll ◽

Ashish Goel

Keyword(s):

Lower Bounds ◽

Excluded Minor ◽

Graph Families

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Densities of Minor-Closed Graph Families

The Electronic Journal of Combinatorics ◽

10.37236/408 ◽

2010 ◽

Vol 17 (1) ◽

Cited By ~ 6

Author(s):

David Eppstein

Keyword(s):

Order Type ◽

Closed Graph ◽

Simple Graphs ◽

Minimal Graphs ◽

Vertex Graph ◽

Graph Families ◽

A Minor ◽

Cluster Points

We define the limiting density of a minor-closed family of simple graphs $\mathcal{F}$ to be the smallest number $k$ such that every $n$-vertex graph in $\mathcal{F}$ has at most $kn(1+o(1))$ edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, well-ordered, and closed; its order type is at least $\omega^\omega$. It is the closure of the set of densities of density-minimal graphs, graphs for which no minor has a greater ratio of edges to vertices. By analyzing density-minimal graphs of low densities, we find all limiting densities up to the first two cluster points of the set of limiting densities, $1$ and $3/2$. For multigraphs, the only possible limiting densities are the integers and the superparticular ratios $i/(i+1)$.

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